Fuzzy Cognitive Map-Based Analysis for Optimizing Watermelon Production Management

Watermelon is a highly sought-after fruit, known for its sweet and refreshing taste. In recent years, its cultivation has gained popularity, particularly in coastal regions, due to the substantial profit margins it offers to farmers. As a result, watermelon production has played a significant role in improving the socio-economic conditions of many individuals.

According to the Bangladesh Bureau of Statistics, the total watermelon yield in the 2019-20 fiscal year reached 254,814 metric tons, cultivated on 30,262 acres of land, with an average production of 8,420 kg per acre. In the following fiscal year, 345,955.44 metric tons were produced on 40,860.73 acres of land, yielding 8,466.70 kg per acre. The Department of Agricultural Marketing (DAM) reported that the average production cost per kg of watermelon in the 2020-2021 fiscal year was Tk. 8.68, with a per-acre cost of Tk. 92,869. Considering a selling price of Tk. 26 per kg, an acre of watermelon cultivation generated a substantial profit of Tk. 188,755 for farmers.

Agricultural production is inherently influenced by various factors, including seasonal and climatic variations, soil type, moisture content, and soil composition. The complex, nonlinear relationships among these factors create challenges in accurately predicting crop yields. To address this issue, Kosko (1994) introduced the Fuzzy Cognitive Map (FCM) technique, which overcomes the limitations of traditional statistical models and the subjectivity of human expertise by aggregating expert opinions to determine the strength of relationships among factors influencing yield [1].

FCMs have been demonstrated as an effective modeling method in precision agriculture, with learning methodologies based on algorithms developed for induced fuzzy cognitive maps to adapt cause-and-effect relationships in FCM models [2]. This approach enhances the effectiveness and strength of FCMs by updating the initial knowledge of human experts and integrating their structural knowledge [3]. In light of these advancements, the present study aims to apply FCM technology, coupled with its learning capabilities, to model watermelon production levels under various agro-climatic conditions. This novel application of FCMs in the context of watermelon cultivation has the potential to contribute valuable insights for optimizing production management and precision agriculture practices.

FCMs have their foundation in graph theory, which was introduced by Euler in 1736. Directed graphs (digraphs) were later employed to investigate real-world structures. Signed digraphs were used to represent information sources, with the term "cognitive map" describing the graphed causal links between variables. Kosko first coined the term "fuzzy cognitive map" to characterize a cognitive map model with two distinguishing features: (a) fuzzy causal relationships between nodes, and (b) dynamic feedback, where changes in one node affect other nodes, which can, in turn, influence the initiating node. The FCM structure is analogous to a recurrent artificial neural network, where concepts are represented by neurons and causal relationships by weighted links connecting the neurons. The concepts encompass the system's characteristics, properties, qualities, and senses. The causal relationships between the concepts are demonstrated through their connections, highlighting the cause-and-effect influence of one FCM concept on the others. These weighted connections indicate the direction and strength of the influence that one concept exerts on others. Figure 1 illustrates the graphical representation of a Fuzzy Cognitive Map [4].

The connection strength between two nodes, $c_i$ and $c_j$, is represented by $w_{i j}$, which takes a value in the range of -1 to 1. There are three possible types of causal relationships among concepts:

•$w_{j i}>0$, which indicates a positive causality between concepts $C_j$ and $C_i$. An increase in the value of $C_j$ leads to an increase in the value of $C_i$, and a decrease in the value of $C_j$ results in a decrease in the value of $C_i$.

•$w_{j i}<0$, which indicates a negative causality between concepts $C_j$ and $C_i$. An increase in the value of $C_j$ leads to a decrease in the value of $C_i$, and a decrease in the value of $C_j$ results in an increase in the value of $C_i$.

•$w_{j i}=0$, which indicates no relationship between $C_j$ and $C_i$.

Definition 2.2.1

A fuzzy cognitive map is associated with directed graphs, where concepts are represented as nodes and the edges represent their causality. If an increase (or decrease) in one concept/node results in an increase (or decrease) in another one, the value is assigned as 1; otherwise, the value would be -1. If no connection exists between concepts, the value would be 0. Consider an FCM with $n$ nodes $C_1, C_2, C_3, \ldots, C_n$, and the directed graph is drawn with weights $w_{i j} \in\{1,-1,0\}$. A square matrix defined by $E=\left(w_{i j}\right)$ is said to be an adjacency matrix of the FCM, where $w_{i j}$ represents the weight of the directed edge $C_i C_j$.

Definition 2.2.2

For an FCM with $n$ nodes $C_1, C_2, C_3, \ldots, C_n$, a vector $A=\left\{a_1, a_2, a_3, \ldots, a_n\right\}$ is called an instantaneous state neutrosophic vector with two states of the nodes at an instant, where the state of the node is defined as:

Definition 2.2.3

Consider an FCM with $n$ nodes $C_1, C_2, C_3, \ldots, C_n$ and the edges of the FCM forming a directed cycle as $\overrightarrow{C_1 C_2}, \overrightarrow{C_2 C_3}, \ldots, \overrightarrow{C_l C_J}$. When an FCM contains a directed cycle, it is classified as cyclic. In contrast, if an FCM lacks a directed cycle, it is considered acyclic. An FCM that includes a cycle is known as a dynamical system. If $C_i$ is activated, and the causality circulates continuously through the cycle's edges, such as $\overrightarrow{C_1 C_2}, \overrightarrow{C_2 C_3}, \ldots, \overrightarrow{C_{l-1} C_l}$, the resulting equilibrium state is referred to as the hidden pattern. When the equilibrium state of a dynamical system corresponds to a unique state vector, it is termed a fixed point. For instance, if the FCM reaches stability with $C_1$ and $C_n$ ON, meaning the state vector remains as $(1,0,0, \ldots, 1)$, this state vector is called a fixed point.

Definition 2.2.4

For any FCM, if the state vector follows the pattern $A_1 \rightarrow A_2 \rightarrow \cdots A_i \rightarrow A_1$, the equilibrium is called a limit cycle of the FCM.

Definition 2.2.5

Consider a vector $A=\left(a_1, a_2, \ldots, a_n\right)$ that passes through a dynamical system $B$. After thresholding and updating the vector $A B$, we obtain the resulting vector $C=\left(b_1, b_2, \ldots, b_n\right)$, denoted as $\left(a_1, a_2, \ldots, a_n\right) \hookrightarrow\left(b_1, b_2, \ldots, b_n\right)$. The symbol $\hookrightarrow$ depicts that the resulting vector has been thresholded and updated.

The Induced Fuzzy Cognitive Map (IFCM) focuses on the FCM algorithm, which generates an optimistic solution from unsupervised data [5]. The following steps must be taken to obtain such an optimistic solution for a problem using unsupervised data:

Three agricultural domain experts helped identify the 15 principles that constitute the FCM model for managing watermelon production. These principles represent different soil types, seasonal and weather factors that experts believe influence watermelon production. The decision concept (DC), in this case, the yield category for watermelons, is one of the concepts and is entirely dependent on the 15 concepts listed in the description provided in the Table 1.

The directed graph of concepts influencing the watermelon production, see Figure 2.

The watermelon production factors (pH, EC, OM, N, P, K, S, Zn, B, Ca, Mg, Cu, Fe, Mn, Temp) are used as the row and column headers, respectively, to create the connection or adjacency matrix M. Values are assigned as 1 if there is a relationship between nodes and 0 if there isn't. To investigate the problem, we will use three different scenarios. In case 1, the active concepts are pH, EC, OM, N, and Production (Table 2, Table 3, Table 4, Table 5, Table 6, and Table 7). In case 2, the activated concepts are K, S, Zn, B, Ca, and Production (Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, and Table 14). Finally, in case 3, the activated concepts are Mg, Cu, Fe, Mn, Temp, and Production (Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, and Table 21).

Assume that the concept pH is active and all other nodes are in an inactive mode for individuals in our directed graph of watermelon production. Thus, pH equals 1. This is represented by the vector $C_1$ = (1,0,0,0,0). The processing according to the algorithm in the Induced Fuzzy Cognitive Map is given in Table 2.

Various measures can be employed for evaluating the FCM; however, in this study, the focus is on centrality measures [6], [7], [8]. Degree centrality and closeness centrality are utilized to present the findings of the analyses in this section. In a given weighted and directed graph, the degree centrality of each node or concept is determined by summing the absolute weights of its outgoing and incoming edges. This metric indicates the extent to which a concept node in an FCM influences other concept nodes within the network. Closeness centrality is defined as the inverse of the sum of the lengths of the shortest paths connecting a node to all other nodes. This measure demonstrates the rate at which a concept section node affects other nodes in the FCM [9], [10], [11], [12], [13], [14], [15].

Degree centrality of a node is determined by the number of edges connected to it. To obtain the standardized score, each value is divided by n-1 (where, n represents the number of nodes). This measure illustrates the extent to which a concept node in an FCM influences other concept nodes within the system (Table 22 and Table 23).

In cases where two nodes are disconnected, the distance between them is considered infinite, as no direct or indirect path connects them. Consequently, the total distances between any two nodes are infinite if at least one is unreachable by the others. Thus, the closeness measure is only applied to the largest node component (measured intra-component). In the subsequent table, F denotes farness, and C represents closeness.

This study has identified the most influential factors affecting watermelon production in Bangladesh, including soil pH, electric conductivity, organic matter, N, P, K, Ca, Mg, and temperature. Moreover, it has been found that humidity and seasonal rainfall play significant roles in ensuring optimal yield. With the assistance of experts in the field, a qualitative description of each concept, their numerical range, a directed graph of concepts influencing watermelon production, and a fuzzy rule base of the concepts have been established.

Future research could extend this work by incorporating nonlinear Hebbian learning algorithms (NHL) and data-driven nonlinear Hebbian learning algorithms (DDNHL) using expert opinions and historical records.